Two cards are drawn successively one by one with out replacement from a pack of cards. The mean of number of kings is

Options

A. `(1)/(13)`
B. `(2)/(13)`
C. `(3)/(13)`
D. `(4)/(13)`

Answer

To solve the problem of finding the mean number of kings when two cards are drawn successively without replacement from a pack of cards, we can follow these steps:

### Step 1: Define the Random Variable
Let \( X \) be the random variable representing the number of kings drawn. The possible values of \( X \) are 0, 1, or 2.

### Step 2: Calculate the Probability for Each Case
We need to calculate the probabilities for each case: \( P(X = 0) \), \( P(X = 1) \), and \( P(X = 2) \).

#### Case 1: \( P(X = 0) \) (No kings drawn)
To find this probability, we can choose 2 cards from the 48 non-king cards:
\[ P(X = 0) = \frac{\binom{48}{2}}{\binom{52}{2}} = \frac{\frac{48 \times 47}{2}}{\frac{52 \times 51}{2}} = \frac{48 \times 47}{52 \times 51} \]
Calculating the values:
\[ = \frac{2256}{2652} = \frac{188}{221} \]

#### Case 2: \( P(X = 1) \) (One king drawn)
To find this probability, we can choose 1 king from the 4 kings and 1 non-king from the 48 non-kings:
\[ P(X = 1) = \frac{\binom{4}{1} \cdot \binom{48}{1}}{\binom{52}{2}} = \frac{4 \times 48}{52 \times 51} \]
Calculating the values:
\[ = \frac{192}{2652} = \frac{32}{221} \]

#### Case 3: \( P(X = 2) \) (Two kings drawn)
To find this probability, we can choose 2 kings from the 4 kings:
\[ P(X = 2) = \frac{\binom{4}{2}}{\binom{52}{2}} = \frac{6}{2652} = \frac{1}{442} \]

### Step 3: Summarize the Probabilities
Now we summarize the probabilities:
- \( P(X = 0) = \frac{188}{221} \)
- \( P(X = 1) = \frac{32}{221} \)
- \( P(X = 2) = \frac{1}{221} \)

### Step 4: Calculate the Mean
The mean \( E(X) \) is calculated using the formula:
\[ E(X) = \sum (X \cdot P(X)) \]
Calculating:
\[ E(X) = 0 \cdot \frac{188}{221} + 1 \cdot \frac{32}{221} + 2 \cdot \frac{1}{221} \]
\[ = 0 + \frac{32}{221} + \frac{2}{221} = \frac{34}{221} \]

### Step 5: Simplify the Mean
To simplify \( \frac{34}{221} \):
\[ \frac{34}{221} = \frac{2}{13} \]

### Final Answer
Thus, the mean number of kings drawn when two cards are drawn is:
\[ \frac{2}{13} \]